we-solve (beta)

Phys-70 · physics · mathematics · Physics

A uniform bound on excess electrons in full Coulomb molecules

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the full nonrelativistic Coulomb Hamiltonian with two electron spin states, M fixed nuclei of arbitrary positions and charges $Z_j\geq 1$ and total $Z=sum$ $Z_j$, is strict n-electron binding possible only for $n\leq Z+CM$ for one universal constant C independent of charges and positions? Also ask that adding electrons above that threshold does not lower the ground energy.

Short English statement

For the full nonrelativistic Coulomb Hamiltonian with two electron spin states, M fixed nuclei of arbitrary positions and charges $Z_j\geq 1$ and total $Z=sum$ $Z_j$, is strict n-electron binding possible only for $n\leq Z+CM$ for one universal constant C independent of charges and positions? Also ask that adding electrons above that threshold does not lower the ground energy.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-70-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-70

Coulomb molecules are claimed to bind at most $Z+CM$ electrons, and neutral atoms to have universally bounded first ionization energy and half-electron exterior radius. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A uniform bound on excess electrons in full Coulomb molecules

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Coulomb molecules are claimed to bind at most $Z+CM$ electrons, and neutral atoms to have universally bounded first ionization energy and half-electron exterior radius.

Mathematical statement of the claimed result

Result 1.
For $M$ fixed nuclei at distinct positions $R_j\in\mathbb R^3$, charges $z_j\ge1$, and total $Z=\sum_jz_j$, use the antisymmetric spin-two electron Hamiltonian \[ H_n=\sum_{i=1}^n\left(-\tfrac12\Delta_i-\sum_j\frac{z_j}{|x_i-R_j|}\right) +\sum_{i<j}\frac1{|x_i-x_j|},\qquad E_n=\inf\sigma(H_n),\quad E_0=0. \] The claimed constant $C<\infty$ is universal over charges and positions, and obeys \[ E_n<E_{n-1}\quad\Longrightarrow\quad n\le Z+CM, \qquad n>Z+CM\quad\Longrightarrow\quad E_n=E_{n-1}. \] For a single nucleus the strict-binding bound becomes $n\le Z+C$, including real $Z\ge1$.

Outline of the argument

Ground states are compared across particle sectors using common localization and screening fields. Local count bounds, changes of observation law, and estimates in nuclear voids control excess particles; deterministic comparison fields give both sides of the neutral-atom energy and radius bounds.

Formal verification

For family 263, the documented entries give the large-$m$ ionization-energy coefficient and the generalized neutral outer-radius asymptotics. Fixed-$m$ limits and the general uniform excess-charge manuscript are not selected. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →